Understanding Units in Physical Quantities

In physics and measurements, a unit is a standard amount of a physical quantity, like length, mass, or time, used as a basis for measurement. Most physical quantities are expressed with both a numerical value and a unit (e.g., 10 meters, 5 kilograms). However, some quantities are dimensionless, meaning they are ratios of quantities with the same units. When you divide a quantity by another quantity that has the exact same unit, the units cancel out, leaving a pure number with no unit.

Analyzing Options for Quantities with No Unit

Let's examine each given option to determine which physical quantity has no unit.

Density Explained with Units

Density is a measure of how much mass is contained within a given volume. It is defined as mass per unit volume. The formula for density is:

\(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\)

Units for mass include kilograms (kg) or grams (g). Units for volume include cubic meters (\(\text{m}^3\)) or cubic centimeters (\(\text{cm}^3\)). Therefore, the standard unit for density in the SI system is kilograms per cubic meter (\(\text{kg/m}^3\)) or grams per cubic centimeter (\(\text{g/cm}^3\)). Density clearly has a unit.

Relative Density and Why it Has No Unit

Relative density, also known as specific gravity, is the ratio of the density of a substance to the density of a reference substance. For liquids and solids, the reference substance is typically water at a specific temperature and pressure (often 4°C and standard atmospheric pressure). The formula for relative density is:

\(\text{Relative Density} = \frac{\text{Density of Substance}}{\text{Density of Reference Substance}}\)

Since both the numerator (Density of Substance) and the denominator (Density of Reference Substance) have the same units (e.g., \(\text{kg/m}^3\) or \(\text{g/cm}^3\)), these units cancel out in the division:

\(\text{Relative Density} = \frac{\text{Unit of Density}}{\text{Unit of Density}} = \frac{\cancel{\text{Unit}}}{\cancel{\text{Unit}}} = \text{No Unit}\)

Thus, relative density is a dimensionless quantity. It is expressed as a pure number.

Displacement and Its Units

Displacement is the shortest distance between the initial and final positions of an object, along with direction. It is a vector quantity representing a change in position. Since displacement measures a distance, its unit is a unit of length. The standard SI unit for displacement is the meter (m). Other units include kilometers (km), centimeters (cm), feet (ft), etc. Displacement definitely has a unit.

Pressure Definition and Units

Pressure is defined as the force applied perpendicular to the surface of an object per unit area over which that force is distributed. The formula for pressure is:

\(\text{Pressure} = \frac{\text{Force}}{\text{Area}}\)

Units for force include Newtons (N). Units for area include square meters (\(\text{m}^2\)). Therefore, the standard SI unit for pressure is Newtons per square meter (\(\text{N/m}^2\)), which is also called the Pascal (Pa). Pressure has a unit.

Comparing Quantities and Their Units

Let's summarize the quantities and their units:

Based on this analysis, relative density is the only quantity among the options that does not have a unit. It is a ratio of two quantities with the same units, resulting in a dimensionless value.

Revision Table: Units of Physical Quantities

Revisiting the key quantities and their units helps reinforce understanding for exam preparation.

Additional Information on Dimensionless Quantities

Relative density is one example of a dimensionless quantity. There are many others in physics and engineering. Understanding why a quantity is dimensionless often involves looking at how it is defined or calculated. If it's a ratio of two quantities with the same physical dimension (and thus the same units), the units cancel out, resulting in a dimensionless number.

Other examples of dimensionless quantities include:

• Strain (ratio of change in length to original length)
• Refractive Index (ratio of speed of light in vacuum to speed of light in medium)
• Angles (measured in radians or degrees, which are essentially ratios of lengths)
• Reynolds Number (a ratio used in fluid dynamics)

Dimensionless quantities are important because their value is independent of the units system used (e.g., SI, CGS, Imperial), making them universally comparable.